Hello, welcome back. In the previous lecture we started with this argument that suppose we know what the optimal risky portfolio looks like. Given that assumption, how do we choose how much do you miss in the risky portfolio and the risky asset. And the answer to that question required us to divide the capital allocation line. Specifically the capital allocation line gives us the risk of return combinations portfolios that we can construct from that optimal risky portfolio, and a risk free asset. We also define the slope of the capital allocation line as the sharp ratio, which gives us the excess reward or the exist return that the risky portfolio provides us per unit of risk. So what I want to do next is plug in some numbers with some data to see what results we can obtain with these concepts. So let's suppose that the risky portfolio that we can invest in is the US equity market. And let's suppose that the risk free rate. The risk free assets Is represented by the T-bills, Treasury bills. Now, let's suppose that the average annual return and volatility of the US equity market, I'm going to denote that as rus, was around 11.9%, this is the same data that we used before. So the average annual return on the US equity market is 11.9%. So I'm going to use that as the expected return. And the volatility of the US equity market over this period was 19.15%. I'll state the risk free rate to be equal to 1%. Now, recall that we can write down the expression for the capital allocation line as the expected return on a portfolio that we construct from these two assets, being equal to the risk free rates plus the excess return on the US equity market minus the risk free rate. Divided by the volatility of the US equity market, that's the Sharpe Ratio range basically times volatility of this portfolio. In other words we can write it as 1% plus the Sharpe Ratio times the volatility of our portfolio, what is the Sharpe Ratio here? The Sharpe Ratio is, of course, going to be 0.119- 0.01, the risk-free rate divided by the volatility, 0.1915 which gives us 0.569. Again, what is this? This is the reward, to volatility ratio. It's the excess return that we're getting from investing in the risky asset, per unit of risk. So let me plot the capital allocation line. Remember, it lives in the mean volatility space. There is the risk-free rate. I'm not going to draw the scale but here is the US Equity Market Volatility, 19.15%, there is the average annual return, 11.9. So given these data, the capital allocation line would look something like this. And again, the slope of course is what was the Sharpe Ratio, given the data we have, it's going to equal to 0.569. And what is this? Well, this is the reward, the mean excess return per unit of risk that US equity markets provides given this data. Now, of course, the question is, where along the Capital Allocation Line would you want to be? Well, last time we saw that that depends on our preferences. It depends on our risky attitude. And we saw, we looked at the analytical solution to that problem. Maximize expected utility by using mean variance utility, and we found the optimal solution, which was basically the excess return on the US equity market, divided by the risk aversion coefficient times the volatility of US equity market,. So now given that we have data, assumptions for these parameters, we can actually plug in different, risk aversion coefficients and find out what the optimal weight would be. So pick a risk aversion coefficient for yourself and given these data you can solve for what the optimal weight you would invest in the US equity market given these assumptions. So in this table, we have the solutions to this optimal allocation problem for various different levels of risk aversion. As you can see, as risk aversion increases, the weight invested in the equity market, the US equity market is decreasing. The third column gives us the weight in the risk free asset. Which is the opposite of the column two. The counter part of column two and the last two columns gives us the expected return, and volatility that we would achieve from these allocations. Now, before we finish, let me point out one more thing. You notice that when we assume a risk aversion coefficient of 1, the weight in the US equity market is 2.9 so 297%. And the weight in there's three asset is at negative percent. It's negative 197%. What does it mean to have a negative weight in the risky assets? How would you make sense of these weights that are greater than 100%? Well, this is about leverage. So how do you construct a portfolio that is more than 100% in the US equity market? Well, so let me illustrate that. So here is my capitalization line. So let me pick two portfolios now on this capital allocation line. Let's pick portfolio k and portfolio l, well, notice that portfolio k is constructed by, Going long in the US equity market. A long in the risk free asset. So for example you invest 50% of your wealth in the executive market and 50% of your wealth in treasury bills. What about portfolio l, or portfolio l on the other hand, is on this higher part of the cathal equation line, which means that you're in fact, investing more than 100% in the equity market. Well, how do you do that? Well, you are borrowing or shorting the risk free assets and investing more than your wealth in the risky asset or the US equity in the states algorithm. So this region corresponds to lending region for if you will, because you're buying treasury bills and this region is corresponds to borrowing region. Why? Because you're borrowing in addition to your wealth and investing more than your wealth in the risky portfolio. And that's what we call leverage. So in this lecture, we use some data to solve the optimal allocation problem for different levels of risk aversion. We also talked about how you would lever up to construct a portfolio that is more than 100% invested in the risky portfolio.